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Properties

Label 20.24.1.b.2
Level $20$
Index $24$
Genus $1$
Analytic rank $0$
Cusps $4$
$\Q$-cusps $2$

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Invariants

Level: $20$ $\SL_2$-level: $10$ Newform level: $80$
Index: $24$ $\PSL_2$-index:$24$
Genus: $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (of which $2$ are rational) Cusp widths $2^{2}\cdot10^{2}$ Cusp orbits $1^{2}\cdot2$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 10D1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 20.24.1.6

Level structure

$\GL_2(\Z/20\Z)$-generators: $\begin{bmatrix}7&12\\12&15\end{bmatrix}$, $\begin{bmatrix}10&7\\7&13\end{bmatrix}$, $\begin{bmatrix}10&13\\13&2\end{bmatrix}$, $\begin{bmatrix}13&10\\19&1\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 20.48.1-20.b.2.1, 20.48.1-20.b.2.2, 20.48.1-20.b.2.3, 20.48.1-20.b.2.4, 20.48.1-20.b.2.5, 20.48.1-20.b.2.6, 40.48.1-20.b.2.1, 40.48.1-20.b.2.2, 40.48.1-20.b.2.3, 40.48.1-20.b.2.4, 40.48.1-20.b.2.5, 40.48.1-20.b.2.6, 60.48.1-20.b.2.1, 60.48.1-20.b.2.2, 60.48.1-20.b.2.3, 60.48.1-20.b.2.4, 60.48.1-20.b.2.5, 60.48.1-20.b.2.6, 120.48.1-20.b.2.1, 120.48.1-20.b.2.2, 120.48.1-20.b.2.3, 120.48.1-20.b.2.4, 120.48.1-20.b.2.5, 120.48.1-20.b.2.6, 140.48.1-20.b.2.1, 140.48.1-20.b.2.2, 140.48.1-20.b.2.3, 140.48.1-20.b.2.4, 140.48.1-20.b.2.5, 140.48.1-20.b.2.6, 220.48.1-20.b.2.1, 220.48.1-20.b.2.2, 220.48.1-20.b.2.3, 220.48.1-20.b.2.4, 220.48.1-20.b.2.5, 220.48.1-20.b.2.6, 260.48.1-20.b.2.1, 260.48.1-20.b.2.2, 260.48.1-20.b.2.3, 260.48.1-20.b.2.4, 260.48.1-20.b.2.5, 260.48.1-20.b.2.6, 280.48.1-20.b.2.1, 280.48.1-20.b.2.2, 280.48.1-20.b.2.3, 280.48.1-20.b.2.4, 280.48.1-20.b.2.5, 280.48.1-20.b.2.6
Cyclic 20-isogeny field degree: $6$
Cyclic 20-torsion field degree: $48$
Full 20-torsion field degree: $1920$

Jacobian

Conductor: $2^{4}\cdot5$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 80.2.a.b

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - x^{2} - 41x + 116 $
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Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Weierstrass model
$(4:0:1)$, $(0:1:0)$

Maps to other modular curves

$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{728x^{2}y^{6}-615053640x^{2}y^{4}z^{2}+1564941405544x^{2}y^{2}z^{4}-457916259765624x^{2}z^{6}-186560xy^{6}z+12246636672xy^{4}z^{3}-16353759767744xy^{2}z^{5}+3704620361328128xz^{7}-y^{8}+18969452y^{6}z^{2}-150229845670y^{4}z^{4}+81666259786092y^{2}z^{6}-7491821289062529z^{8}}{y^{2}(x^{2}y^{4}-22x^{2}y^{2}z^{2}-x^{2}z^{4}+14xy^{4}z-65xy^{2}z^{3}-3xz^{5}+47y^{4}z^{2}+645y^{2}z^{4}+29z^{6})}$

Modular covers

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Cover information

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The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
4.2.0.a.1 $4$ $12$ $12$ $0$ $0$ full Jacobian
5.12.0.a.2 $5$ $2$ $2$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
5.12.0.a.2 $5$ $2$ $2$ $0$ $0$ full Jacobian
20.12.0.p.2 $20$ $2$ $2$ $0$ $0$ full Jacobian
20.12.1.a.1 $20$ $2$ $2$ $1$ $0$ dimension zero

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
20.48.3.b.1 $20$ $2$ $2$ $3$ $0$ $2$
20.48.3.d.2 $20$ $2$ $2$ $3$ $0$ $2$
20.72.1.b.2 $20$ $3$ $3$ $1$ $0$ dimension zero
20.96.5.c.2 $20$ $4$ $4$ $5$ $0$ $1^{2}\cdot2$
20.120.5.p.1 $20$ $5$ $5$ $5$ $0$ $1^{2}\cdot2$
40.48.3.b.1 $40$ $2$ $2$ $3$ $0$ $2$
40.48.3.d.2 $40$ $2$ $2$ $3$ $0$ $2$
60.48.3.e.2 $60$ $2$ $2$ $3$ $0$ $2$
60.48.3.g.1 $60$ $2$ $2$ $3$ $0$ $2$
60.72.5.b.1 $60$ $3$ $3$ $5$ $0$ $1^{2}\cdot2$
60.96.5.b.1 $60$ $4$ $4$ $5$ $0$ $1^{2}\cdot2$
100.120.5.b.2 $100$ $5$ $5$ $5$ $?$ not computed
120.48.3.f.1 $120$ $2$ $2$ $3$ $?$ not computed
120.48.3.h.2 $120$ $2$ $2$ $3$ $?$ not computed
140.48.3.b.2 $140$ $2$ $2$ $3$ $?$ not computed
140.48.3.d.2 $140$ $2$ $2$ $3$ $?$ not computed
140.192.13.b.1 $140$ $8$ $8$ $13$ $?$ not computed
220.48.3.b.2 $220$ $2$ $2$ $3$ $?$ not computed
220.48.3.d.2 $220$ $2$ $2$ $3$ $?$ not computed
220.288.21.b.1 $220$ $12$ $12$ $21$ $?$ not computed
260.48.3.b.2 $260$ $2$ $2$ $3$ $?$ not computed
260.48.3.d.1 $260$ $2$ $2$ $3$ $?$ not computed
280.48.3.b.2 $280$ $2$ $2$ $3$ $?$ not computed
280.48.3.d.2 $280$ $2$ $2$ $3$ $?$ not computed